LOCAL SOLUTION OF CARRIER’S EQUATION
IN A NONCYLINDRICAL DOMAIN
TANIA NUNES RABELLO AND MARIA CRISTINA DE CAMPOS VIEIRA
Received 29 April 2003
We study Carrier’s equation in a noncylindrical domain. We use the penalty method
combined with Faedo-Galerkin and compactness arguments. We obtain results of the
existence and uniqueness of the local solution.
1. Introduction
The wave equation of the type
∂2 u
− P0 + P1
∂t 2
β
α
2
∂u
∂2 u
=0
(t) dx
∂x
∂x2
(1.1)
is a model for small vibrations of an elastic string fixed at α, β with P0 and P1 constants,
and was investigated by Kirchhoff [4] and Carrier [1]. By approximations on the above
equation, Carrier obtained the model
∂2 u
− P0 + P1
∂t 2
β
α
2
u(t) dx
∂2 u
= 0,
∂x2
(1.2)
which is known in the literature as Carrier’s equation. Medeiros et al. [7, 8] studied the
problem of small vibrations of an elastic string with moving boundary.
In this work, we study the following generalization of (1.2):
∂2 u − M x,t,
∂t 2
β(t)
α(t)
u(t) dx
u=0
u(x,0) = u0 (x),
2
∂2 u
=f
∂x2
on ,
ut (x,0) = u1 (x),
in Q,
in α(0), β(0) ,
where
=
(i) Q
]α(t),β(t)[×{t } is a noncylindrical domain,
0<t<T
(ii) = 0<t<T {(α(t),t),(β(t),t)} is the lateral boundary of Q.
Copyright © 2005 Hindawi Publishing Corporation
Journal of Inequalities and Applications 2005:3 (2005) 303–317
DOI: 10.1155/JIA.2005.303
(1.3a)
(1.3b)
(1.3c)
304
Local solution of Carrier’s equation in a noncylindrical domain
We use the penalty method as in Ebihara [2] combined with the Faedo-Galerkin
method and compactness arguments. We obtain results of the existence and uniqueness
of the local solution. This method was used by several authors, for example, Ebihara et
al. [3], Pereira et al. [9]. We use some ideas contained in [7, 8] to obtain the necessary
estimates.
2. Notations, assumptions, and main results
We use the notations contained in Lions and Magenes [6] and Lions [5] for the spaces
L p (Ω), W m,p (Ω), 1 ≤ p ≤ ∞, m ∈ N. In particular, for p = 2, m = 1, and Ω = (0,1), we
consider L2 (0,1) and H01 (0,1) with the scalar product and norms represented by (·, ·),
| · |, ((·, ·)), and · , respectively, and given by
(u,v) =
(u,v) =
1
1
0
0
u(x)v(x)dx,
|u|2 =
1
0
2
u(x) dx,
2
1
du
(x) dx.
u =
dx
du dv
(x) (x)dx,
dx
dx
(2.1)
2
0
We consider the following assumptions about M:
∞
1,
∞ × (0,L));
(M1) M ∈ Lloc ([0, ∞),W (Q)) and for each L > 0, ∂M/∂λ
∈ L ∞ (Q
(M2) M, ∂M/∂x, ∂M/∂λ, ∂M/∂t are continuous with respect to λ, a.e. (x,t) ∈ Q;
and
(M3) there exists a real number m0 > 0 such that M(x,t,λ)
≥ m0 for all (x,t) ∈ Q
λ ≥ 0.
we consider the following assumptions:
With respect to the noncylindrical domain Q
(H1) α, β ∈ C 3 ([0,T]), α(t) < β(t), for all t ∈ [0,T];
β (t) > 0, for all t ∈ (0,T], and α (0) = 0 = β (0);
(H2) α (t) < 0,
(H3) γ (t) ≤ m0 /2, for all t ∈ [0,T], where γ(t) = β(t) − α(t);
(H4) |α (t) + γ (t)y | ≤ |α (t) + γ (t)y |2 /γ(t), for all t ∈ [0,T], y ∈ [0,1].
the point (y,t) of R2 , with y = (x − α(t))/γ(t),
We observe that when (x,t) varies in Q
→ Q by
varies in Q = (0,1) × (0,T), so we define the diffeomorphism τ : Q
τ(x,t) = (y,t),
where y =
x − α(t)
.
γ(t)
(2.2)
× [0, ∞) → Q × [0, ∞) defined by τ(x,t,λ) =
We also consider the diffeomorphism τ : Q
(y,t,λ) with y as above.
Denoting
v(y,t) = u ◦ τ −1 (y,t),
◦ (τ)−1 (y,t,λ),
M(y,t,λ) = M
g(y,t) = f ◦ τ
−1
(y,t),
(2.3)
T. N. Rabello and M. C. de Campos Vieira 305
v0 (y) = u0 (γ0 y + α0 ), and v1 = (γ0 y + α0 ), where γ0 = γ(0), α0 = α(0), and defining
b(y,t) = −2
α (t) + γ (t)y
,
γ(t)
α (t) + γ (t)y
α (t) + γ (t)y
− 2γ (t)
,
c(y,t) = −
γ(t)
γ2 (t)
(2.4)
problem (1.3) is transformed into the following cylindrical problem:
2 b2 (y,t) ∂2 v
1
(y,t)
v (y,t) − 2 M y,t,γ(t)v(t) −
γ (t)
4
∂y 2
∂v
∂v
+b(y,t)
(y,t) + c(y,t) (y,t) = g(y,t),
∂y
∂y
v=0
v(y,0) = v0 (y),
on
(2.5a)
in Q,
,
(2.5b)
v (y,0) = v1 (y),
in (0,1),
(2.5c)
where denotes the derivative with respect to t.
If Ωt = (α(t),β(t)) and Ω0 = (α0 ,β0 ), where α0 = α(0) and β0 = β(0), the main result
of this work is given by the following theorem.
satisfy (M1)–(M3) and let α(t), β(t), and γ(t) satisfy (H1)–(H4).
Theorem 2.1. Let M
1
Given u0 ∈ H0 (Ω0 ) ∩ H 2 (Ω0 ), u1 ∈ H01 (Ω0 ), and f ∈ L2 (0,T,H01 (Ωt )) with f ∈ L2
(0,T,L2 (Ωt )), there exist T0 > 0 and a unique u satisfying
u ∈ L∞ 0,T0 ,H01 Ωt ∩ H 2 Ωt ,
u ∈ L∞ 0,T0 ,H 1 Ωt ,
(2.6)
u ∈ L 0,T0 ,L Ωt ,
2
2
and u is the solution of (1.3).
First of all we prove the equivalent result in a cylindrical domain.
Theorem 2.2. Let α(t), β(t), and γ(t) be as in (H1)–(H4) and let M satisfy
1,∞ (Q)) and for each L > 0, ∂M/∂λ ∈ L∞ (Q × (0,L)),
(i) M ∈ L∞
loc ([0, ∞),W
(ii) M,∂M/∂λ, ∂M/∂t are continuous with respect to λ, a.e. (y,t) ∈ Q.
There exists a real number m0 > 0 such that
M(y,t,λ) ≥ m0 ,
∀(y,t) ∈ Q, λ ≥ 0.
(2.7)
Given v0 ∈ H01 (0,1) ∩ H 2 (0,1), v1 ∈ H01 (0,1), and g ∈ L2 ([0,T],H01 (0,1)) with g ∈ L2
([0,T],L2 (0,1)), there exist T0 > 0 and a unique v satisfying
v ∈ L∞ 0,T0 ,H01 (0,1) ∩ H 2 (0,1) ,
∞
v ∈ L 0,T0 ,H01 (0,1) ,
v ∈ L∞ 0,T0 ,L2 (0,1) ,
and v is the solution of (2.5).
(2.8a)
(2.8b)
(2.8c)
306
Local solution of Carrier’s equation in a noncylindrical domain
3. Proof of results
Let (wν )ν∈N be the orthonormal complete set of L2 (0,1) given by the eigenvectors of the
operator −d2 /dx2 , that is,
−
d2
wν = λν wν ,
dx2
wν = 0
on Γ.
(3.1)
We represent by Vm = [w1 ,...,wm ] the subspace of H01 (0,1) ∩ H 2 (0,1) generated by the
m first vectors wν .
We consider F : (0, ∞) → R a function satisfying
F ∈ C 1 (0, ∞),
F (ξ) < 0,
F(ξ) = 1, ∀ξ ≥ 1.
µ0
There exist µ0 , η0 , δ > 0 such that F(ξ) ≥ η0 , ∀ξ ∈ (0,δ].
ξ
∀ξ > 0,
(3.2)
(3.3)
Let K > 0 such that
2
v1 < K.
(3.4)
For each ε > 0, we consider the following penalized problem: for each m ∈ N, let vεm (t) ∈
Vm satisfy
2 b2 (y,t) ∂2 vεm
1
vεm (t) − 2 M y,t,γ(t)vεm (t) −
(t),w
γ (t)
4
∂y 2
2 K − vεm
(t)
∂v
∂v
+ c(y,t) εm (t) + b(y,t) εm (t) + εF
∂y
∂y
ε
vεm (t),w = g(t),w ,
(3.5)
for all w ∈ Vm , with the initial conditions
vεm (0) = v0m −→ v0
in H01 (0,1) ∩ H 2 (0,1),
vεm
(0) = v1m −→ v1
in H01 (0,1).
(3.6a)
(3.6b)
As in [9] we prove that we can apply Carathéodory’s theorem to (3.5)–(3.6b) obtaining
the existence of Tεm > 0 and vεm : [0,Tεm ) → Vεm as a solution of (3.5) satisfying
2
(t) < K,
vεm ∈ C 2 0,Tεm , vεm
∀m ≥ m1 , t ∈ 0,Tεm , for each ε > 0.
(3.7)
T. N. Rabello and M. C. de Campos Vieira 307
Lemma 3.1. There exist constants C0 > 0 and ε0 > 0 such that
v (0) ≤ C0 ,
εm
(3.8)
for all m ≥ m1 , 0 < ε < ε0 .
Proof. From (3.6) the existence of a constant N > 0 follows, such that
v (0) ≤ N,
εm
vεm (0) ≤ N,
vεm (0) ≤ N,
vεm (0) ≤ N,
∀ m ≥ m1 .
(3.9)
From (3.4) there exists ρ > 0 such that
2
v1 < ρ < K,
(3.10)
so
v1m |2 < ρ < K,
∀ m ≥ m1 .
(3.11)
Then, for each 0 < ε < min{1,K − ρ} = ε0 , we have
2
K − v1m > 1,
ε
∀ m ≥ m1 ,
(3.12)
thus, from (3.2),
2 K − v1m F
ε
= 1,
∀ m ≥ m1 .
(3.13)
Let
M0 =
max
[0,1]×[0,γ0
N2]
M(y,0,λ).
(3.14)
We supposed that α (0) = β (0) = 0, then it follows from (H4) that b(y,0) = c(y,0) =
0, so by considering t = 0 and w = vεm
(0) in (3.5), we obtain, from (3.13),
v (0) ≤ 1 M0 + γ2 N + g(0),
εm
0
2
γ0
and the proof of the lemma is complete.
(3.15)
308
Local solution of Carrier’s equation in a noncylindrical domain
Lemma 3.2. Fix ε > 0 and m ≥ m1 . Then
(t)|2 such that limt→Tεm |vεm
(t)|2 < K.
(i) there exists limt→Tεm |vεm
Proof. Taking the derivate with respect to t of (3.5) we obtain
vεm (t) +
2γ (t) vεm (t)2 − 1 M y,t,γ(t)vεm (t)2
y,t,γ(t)
M
γ3 (t)
γ2 (t)
−
γ (t) ∂M vεm (t)2 vεm (t)2 − 2 ∂M y,t,γ(t)vεm (t)2
y,t,γ(t)
γ2 (t) ∂λ
γ(t) ∂λ
∂2 vεm
1
× vεm (t),vεm (t) + b2 (y,t)
(t)
4
∂y 2
+
2 ∂2 vεm
b2 (y,t)
∂v
1
− 2 M y,t,γ(t)vεm (t)
(t) + c (y,t) εm (t)
2
4
γ (t)
∂y
∂y
+ c(y,t) + b(y,t)
2 K − vεm
(t)
× vεm (t), vεm (t) vεm (t) + εF
ε
2 K − vεm
(t)
∂v
(t) + b(y,t) εm (t)−2F ∂y
∂y
ε
∂vεm
vεm (t),w = (g (t), w).
(3.16)
From (3.7) we obtain that
vεm (t)2 < Kεm ,
∀t ∈ 0,Tεm .
(3.17)
Let
C1εm =
sup ess
M(y,t,λ), M (y,t,λ),
[0,1]×[0,T]×[0,γ(T)Kεm ]
∂M
.
(y,t,λ)
∂λ
(3.18)
We remember that
α (t) + γ (t)y ≤ γ (t),
2
α (t) + γ (t)y ≤ α (t) + γ (t)y ,
γ(t)
(3.19)
t ∈ [0,T], y ∈ [0,1],
T. N. Rabello and M. C. de Campos Vieira 309
so we obtain from (H3)
3
1 2
m0 2m0
b (y,t) ≤ 4 γ (t)
≤
,
4
3
γ(t)
γ0
γ (t) 2 2
≤ 2m0 C1εm ,
γ3 (t) M y,t,γ(t) vεm (t)
3
γ0
1
vεm (t)2 ≤ 1 C1εm ,
y,t,γ(t)
M
γ2 (t)
2
γ0
1 ∂M 2 2
γ (t)vεm (t) + 2γ(t) vεm (t), vεm
(t) γ2 (t) ∂λ y,t,γ(t) vεm t)
≤ Kεm C1εm
1
γ02
m0 2
+ K ,
2 γ0
2
2
b (y,t)
2 1
≤ γ (t) + 1 C1εm ≤ 1 m0 + C1εm ,
−
y,t,γ(t)
v
(t)
M
εm
4
2
γ2 (t)
γ(t)
γ2 (t)
2
γ0
c(y,t) + b (y,t) ≤ 7m0 .
2
2γ0
(3.20)
(t) in (3.16) it follows from (3.20) that
Taking w = vεm
2 1 d
v (t)2 + εF K − vεm (t)
εm
2 dt
ε
≤ vεm
(t)
C1εm
2 2
v (t)2 − 2F K − vεm (t)
vεm
(t), vεm
(t)
εm
ε
1
2m0 1
3 + 2 + 2
γ0 γ0
γ0
2
m 2m ∂2 vεm m0
(t)
Kεm + Kεm K + 0 3 0 ∂y 2
2
γ0
γ0
2 ∂ vεm m0 1
+ C2 vεm (t) + 3m0 + 2m0 v (t)
+
+
C
(t)
1εm
∂y 2
εm
2
2
2
2
2γ0 γ0
2γ0
γ0
+ g (t) +
2m0 v (t)v (t),
εm
εm
γ0
(3.21)
where C2 = max[0,1]×[0,T] |c (y,t)|. But in Vm the norms are equivalent, so by using (3.7)
and (3.17) we obtain that there exist constants C2εm > 0 and C3εm > 0 depending on ε and
m such that
2 1 d
v (t)2 + εF K − vεm (t)
εm
2 dt
ε
−2F 2 K − vεm
(t)
ε
v (t)2
εm
vεm
(t), vεm
(t)
2
2 2 ≤ C2εm vεm
(t) + g (t) + C3εm .
(3.22)
310
Local solution of Carrier’s equation in a noncylindrical domain
Using the hypothesis on F and F , Lemma 3.1, and Gronwall’s inequality in (3.22) it
follows that
v (t)2 ≤ C4εm ,
εm
(3.23)
where C4εm > 0 is a constant depending on ε > 0 and m.
(t)|2 exists and
It follows from (3.7) and (3.23) that limt→Tεm |vεm
2
lim vεm
(t) ≤ K.
(3.24)
2
(t) = K.
lim vεm
(3.25)
t →Tεm
Suppose that
t →Tεm
(t) in (3.5), we have
Taking w = vεm
2 1 d
v (t)2 + εF K − vεm (t)
εm
2 dt
ε
1
+
0
1
+
=
1
0
0
v (t)2
εm
2 ∂2 vεm
b2 (y,t)
1
− 2 M y,t,γ(t)vεm (t)
(y,t)vεm
(y,t)d y
4
γ (t)
∂y 2
∂v
c(y,t) εm (y,t)vεm
(y,t)d y +
∂y
1
0
∂v
b(y,t) εm (y,t)vεm
(y,t)d y
∂y
(3.26)
g(y,t)vεm
(y,t)d y,
but
b2 (y,t)
2 1
1 m0
≤ 2
− 2 M y,t,γ(t) vεm (t)
+ C1εm ,
4
γ0 2
γ (t)
c(y,t) ≤ 3 m0 ,
2
γ0 2
b(y,t) ≤ 2m0 ,
γ0
(3.27)
so we have
2 K − vεm
(t)
εF
ε
v (t)2
εm
2
1 m0
∂ vεm
v (y,t)
≤ vεm
(t)vεm
(t) + 2
(y,t)
+ C1εm εm
2
γ0
2
∂y
3 m 2m0 v (t)v (t) + g(t)v (t),
+ 2 0 vεm (t)vεm
(t) +
εm
εm
εm
2
γ
γ0
0
(3.28)
T. N. Rabello and M. C. de Campos Vieira 311
then from (3.23), (3.17), and (3.7) and the equivalence of the norms in Vm there exists a
constant C5εm > 0 such that
2 K − vεm
(t)
εF
ε
2
vεm
(t) ≤ C5εm .
(3.29)
We conclude from (3.25) and (3.29) that there exists δ1 > 0 such that
2 K − vεm
(t)
εF
ε
≤ C5εm
2
,
K
for t ∈ Tεm − δ1 ,Tεm .
(3.30)
But by (3.25) and the hypothesis on F we have
2 K − vεm
(t)
lim εF
t →Tεm
ε
= ∞,
(3.31)
which is a contradiction to (3.30). Then the proof is completed.
From (3.7) and Lemma 3.2(i) we can extend the solution to [0, T] and
vεm (t)2 ≤ K1 e v (t)2 ≤ K,
εm
∀t ∈ [0,T],
(3.32)
where K and K1 are independent of ε > 0 and m ≥ m1 .
Lemma 3.3. There exists a constant K2 > 0 independent of ε > 0 and m ≥ m1 such that
2
2
v (t)2 + ∂ vεm (t) ≤ K2 ,
εm
2
∂y
∀t ∈ [0,T].
(3.33)
Proof. We observe that
1
0
1
0
c(y,t)
∂vεm (y,t) ∂2 vεm
∂v
∂v
(y,t)d y = c(1,t) εm (1,t) εm (1,t)
2
∂y
∂y
∂y
∂y
1
∂v
∂vεm
∂vεm
∂2 vεm
−c(0,t)
(y,t) εm (y,t)d y
(0,t)
(0,t) − c(y,t)
2
∂y
∂y
∂y
∂y
0
1
∂v
∂c
∂v
−
(y,t) εm (y,t) εm (y,t)d y,
∂y
∂y
0 ∂y
∂v
∂2 v 1
b(y,t) εm (y,t) εm
(y,t)d y =
2
∂y
∂y
2
=
∂vεm
1
b(1,t)
(1,t)
2
∂y
1
−
2
1
0
2
1
2
− b(0,t)
∂vεm
∂b
(y,t)
(y,t)
∂y
∂y
2
1
0
∂vεm
(0,t)
∂y
d y,
∂ ∂vεm
b(y,t)
(y,t)
∂y ∂y
2
2
(3.34)
dy
312
Local solution of Carrier’s equation in a noncylindrical domain
/∂y 2 )(y,t) in (3.5) we obtain
so if we take w = −(∂2 vεm
1
2 b2 (y,t)
1 d 1
v (t)2 +
M y,t,γ(t)vεm (t) −
εm
2
2 dt
γ (t)
4
0
1
∂2 vεm
(y,t)
∂y 2
2
dy
2
2 d b2 (y,t)
1
∂2 vεm
− 2 M y,t,γ(t)vεm (t)
(y,t) d y
4
γ (t)
∂y 2
0 dt
∂v
∂v
∂v
∂v
+ c(0,t) εm (0,t) εm (0,t) − c(1,t) εm (1,t) εm (1,t)
∂y
∂y
∂y
∂y
1
1
2
∂v
∂v
∂v
∂c
∂v
(y,t) εm (y,t)d y +
+ c(y,t) εm
(y,t) εm (y,t) εm (y,t)d y
2
∂y
∂y
∂y
∂y
0
0 ∂y
+
1
2
∂vεm
1
+ b(0,t)
(0,t)
2
∂y
2
∂vεm
1
− b(1,t)
(1,t)
2
∂y
2
1
+
2
1
0
∂vεm
∂b
(y,t)
(y,t)
∂y
∂y
2
dy
≤ g(t) vεm
(t).
(3.35)
It follows from (H3) and (H4) that
c(1,t) ∂vεm (1,t) ∂vεm (1,t)
∂y
∂y
≤ 2
γ (t)β (t)
β (t)
+
2
γ (t)
γ(t)
2 ∂vεm
∂vεm
(1,t)
(1,t)
∂y
∂y
2
3 2
2
β (t) ∂vεm (1,t) + 4 γ (t) β (t) + β (t) ∂vεm (1,t)
≤
∂y
γ(t) ∂y
2γ3 (t)
2
3/2
2
2
2
β (t) π +1 ∂vεm (1,t) + 5 m0
∂ vεm (t) ,
≤
3
2
γ(t) ∂y
π
∂y
2γ0 2
c(0,t) ∂vεm (0,t) ∂vεm (0,t)
∂y
∂y
2
2
2
∂vεm
−α (t) 5 m0 3/2 π + 1 2 ∂ vεm ≤
(0,t) + 3
∂y 2 (t) ,
γ(t) ∂y
π
2γ0 2
1
0
2
2
2
d
1
vεm (t)2 − b (y,t) ∂ vεm (y,t) d y
y,t,γ(t)
M
2
2
dt γ (t)
4
∂y
=
1
0
2 2 2γ (t) 1 ∂M y,t,γ(t)vεm (t)
M y,t,γ(t)vεm (t) + 2
3
γ (t)
γ (t) ∂t
2
1 ∂M
2
+ 2
y,t,γ(t)vεm (t) γ (t)vεm (t) + 2γ(t) vεm (t), vεm
(t)
γ (t) ∂λ
2 + 3
α (t) + γ (t)y α (t) + γ (t)y γ(t)− α (t) + γ (t)y γ (t)
γ (t)
−
2 2
∂ vεm
dy
(y,t)
×
2
∂y
T. N. Rabello and M. C. de Campos Vieira 313
≤ M1
1
0
1
0
1
0
2 m0
γ03 2
1/2
1 m0
+ 2
γ0 2
1/2
2
4 m0
K1 + KK1 + 3
γ0
γ0 2
3/2 2
∂ vεm 2
∂y 2 (t) ,
∂v
∂c
2 ∂v
(t),
(y,t) εm (y,t) εm (y,t)d y ≤ 2 m0 vεm (t) vεm
∂y
∂y
∂y
γ0
∂vεm
∂b
(y,t)
(y,t)
∂y
∂y
c(y,t)
2
dy ≤
2 m0
γ0 2
1/2
v (t)2 ,
εm
∂v
∂2 vεm
3 m ∂2 vεm v (t).
(y,t) εm (y,t)d y ≤ 2 0 (t)
εm
2
2
∂y
∂y
γ0 2 ∂y
(3.36)
By substituting the above inequalities in (3.35) we obtain
1 d
v (t)2 +
2 dt εm
1
0
2
2
2
1
vεm (t)2 − b (y,t) ∂ vεm (y,t) d y
y,t,γ(t)
M
γ2 (t)
4
∂y 2
1
2m0 1
+ 2+ 2
γ0 γ0
γ03
≤ M1
+
2
2
2
m0 2m0 m0
∂ vεm K1 K +
K1 +
∂y 2 (t)
2
γ0
γ03
2
3 m0 v (t) ∂ vεm (t) + 2m0 vεm (t) v (t) + 2
∂y 2
εm
εm
2
2
2
γ0
γ0
γ0
2 2
∂ vεm 2 + g(t) v (t)
(t)
∂y 2
εm
2 ∂2 vεm 2
2
1
≤ C3 vεm (t) + (t)
+ 2 g(t) ,
∂y 2
+
5 m0
γ03 2
3/2 m0 v (t)2
2 εm
π +1
π
(3.37)
where
M1 = max
sup ess
[0,T]×[0,1]×[0,γ0 K1 ]
∂
M(y,t,λ), M (t, y,λ), ∂y
∂
M(y,t,λ)
, ∂λ
M(y,t,λ)
.
(3.38)
Integrating inequality (3.37) from 0 to t, the result follows from the convergence of the
initial data, Gronwall’s lemma, (H6), and (2.7).
Lemma 3.4. There exists a constant K3 > 0 independent of ε > 0 and m ≥ m1 such that
v (t) ≤ K3 ,
εm
∀t ∈ [0,T].
(3.39)
Proof. Considering w = vεm
(t) in (3.16), using Lemmas 3.2 and 3.3, and by the same
arguments used in the proof of Lemma 3.2, it follows that there exists a constant C4 > 0
314
Local solution of Carrier’s equation in a noncylindrical domain
independent of ε > 0 and m ≥ m1 such that
2 2 2
v (t)2 −2F K − vεm (t)
vεm
(t), vεm
(t)
εm
1 d
v (t)2 + εF K − vεm (t)
εm
2 dt
ε
1
ε
2
1
vεm (t)2 − b (y,t)
y,t,γ(t)
M
2
1 d
2 dt 0 γ (t)
2
1 1
2
≤ C4 + vεm
(t) + g (t) .
2
2
+
4
2
∂vεm
∂y (y,t) d y
(3.40)
The result follows from this inequality, the convergence of the initial data, and from Gron
wall’s lemma.
3.1. Proof of Theorem 2.2. We observe that
(t)|2 ≤ K/2, then (K − |vεm
(t)|2 )/ε ≥ K/2ε, so from the hypothesis on F we
(i) if |vεm
have
F
2 K − vεm
(t)
ε
= 1,
for ε > 0 such that 2ε < K;
(3.41)
(ii) if |vεm
(t)|2 > K/2, taking w = vεm
(t) in (3.5), we have
2 K − vεm
(t)
εF
ε
1
2
2 1
vεm (t)2 − b (y,t)
v (t)v (t) +
y,t,γ(t)
≤
M
εm
εm
K
γ2 (t)
4
0
× vεm
(t)d y +
1
c(y,t) ∂vεm (y,t)v (y,t)d y
εm
∂y
∂2 vεm
(y,t)
∂y 2
0
1
1
∂vεm
+ b(y,t)
g(y,t)vεm (y,t) d y .
(y,t)vεm (y,t)d y +
∂y
0
0
(3.42)
Using the estimates obtained in the above inequality, there exists a constant K4 > 0 such
that
2 K − vεm
(t)
εF
ε
≤ K4 ,
for ε > 0.
(3.43)
From the above inequalities, there exists a constant K5 > 0 independent of ε > 0 and
m ≥ m1 such that
2 K − vεm
(t)
εF
ε
≤ K4 ,
∀t ∈ [0,T], m ≥ m1 , 0 < ε <
K
.
2
(3.44)
T. N. Rabello and M. C. de Campos Vieira 315
From the lemmas and (3.44) it follows that there exists a subsequence of (vεm ), still
denoted by (vεm ), such that
vεm −→ v ,
vεm −→ v ,
εF
weakly∗ in L∞ 0,T,H01 (0,1) ∩ H 2 (0,1) ,
vεm −→ v,
∗
weakly in L
∗
∞
ε
−→ χ,
(3.45a)
0,T,H01 (0,1) ,
∞
2
(3.45b)
0,T,L (0,1) ,
(3.45c)
weakly∗ in L∞ (0,T).
(3.45d)
weakly in L
2 K − vεm
(t)
From (3.45a)–(3.45c) we conclude that (vεm (t)) and (vεm
(t)) are equicontinuous with
1
2
respect to the norms on H0 (0,1) and L (0,1), respectively. So there exists a subsequence
of (vεm ), still denoted by (vεm ), such that
strongly in H01 (0,1), uniformly in [0,T],
vεm −→ v,
2
vεm −→ v ,
strongly in L (0,1), uniformly in [0,T].
(3.46)
(3.47)
1,∞ (Q)) and for each L > 0, (∂M/∂λ) ∈ L∞ (Q × (0,L)), so
But M ∈ L∞
loc ([0, ∞) ,W
2 2 M t, γ(t)vεm (t) −→ M t, γ(t)v(t)
strongly in L2 (0,1), uniformly in [0,T].
(3.48)
Passing to the limit, when m → ∞ and ε → 0, in (3.5), we obtain
2 b2 (y,t) ∂2 v
∂v
1
v (t) + − 2 M y,t,γ(t)v(t) +
(t) + c(y,t) (t)
2
γ (t)
4
∂y
∂y
∂v
+ b(y,t)
(t) + χ(t)v (t) = g(t) in L2 0,T,L2 (0,1) ,
∂y
(3.49)
and v(0) = v0 and v (0) = v1 .
We have v ∈ C 0 ([0,T], L2 (0,1)) and |v (0)|2 = |v1 |2 < K, so there exists T0 > 0 such
that
2
v (t) < K,
∀t ∈ 0,T0 .
(3.50)
By considering β0 = max[0,T0 ] |v (t)|2 and ρ0 = K − β0 > 0, we have
2
v (t) ≤ β0 = K − ρ0 < K − ρ0 ,
2
∀t ∈ 0,T0 .
(3.51)
From (3.47) there exists m2 ∈ N such that
v (t)2 < ρ0 + v (t)2 < K − ρ0 ,
εm
4
4
∀t ∈ 0,T0 .
(3.52)
Then, for 0 < ε < (ρ0 /4), we have
2
(t)
K − vεm
≥ 1,
ε
m ≥ m2 , t ∈ 0,T0 ,
(3.53)
316
Local solution of Carrier’s equation in a noncylindrical domain
so
2 K − vεm
(t)
εF
ε
= ε,
t ∈ 0,T0 .
(3.54)
Thus by the uniqueness of the limit it follows that χ = 0 in [0,T0 ].
For the uniqueness we consider v and v solutions in [0,T0 ], and w = v − v. So we have
w (t) + −
2 b2 (y,t) ∂2 w
1
(t)
M y,t,γ(t)v(t) +
2
γ (t)
4
∂y 2
2 2 ∂2 v
1
1
(t)
+ 2 M y,t,γ(t)v(t) − 2 M y,t,γ(t)v(t)
γ (t)
γ (t)
∂y 2
+ c(y,t)
(3.55)
∂w
∂w
(t) + b(y,t)
(t) = 0 in L2 0,T,L2 (0,1) .
∂y
∂y
By defining
2 1
1
Ψ(t) = w (t) +
2
2
1
0
2 b2 (y,t)
1
M y,t,γ(t)v(t) −
2
γ (t)
4
∂w
(t)
∂y
2
dy
(3.56)
and multiplying the equation in (3.55) by w , we obtain
Ψ (t) − CΨ(t) ≤ 0,
(3.57)
for some constant C > 0, and since Ψ(0) = 0 we conclude that w = 0 in [0,T0 ].
3.2. Proof of Theorem 2.1. In this section, we study the noncylindrical problem (1.3).
We represent by Ωt and Ω0 the intervals (α(t),β(t)) and (α0 ,β0 ), respectively, and we
denote by | · |0 , · 0 , | · |t , · t the norms in L2 (Ω0 ), H01 (Ω0 ), L2 (Ωt ), H01 (Ωt ), respectively.
We consider the change of variables x = α(t) + γ(t)y, and we define
α(t) + γ(t)y,t,λ ,
M(y,t,λ) = M
v0 (y) = u0 α0 + γ0 y e v1 (y) = u1 α0 + γ0 y ,
(3.58)
g(y,t) = f α(t) + γ(t)y,t .
We can see that we are in the conditions of Theorem 2.2. It follows that there exist T0 >
0 and a unique solution v of problem (2.5) satisfying conditions (2.8). By considering
u(x,t) = v(y,t) with x = α(t) + γ(t)y we verify that u is the solution of Theorem 2.1.
The regularity and uniqueness of u claimed in Theorem 2.1 are obtained by the regularity and uniqueness of the solution v of Theorem 2.2.
Acknowledgment
We are very grateful to Dr. L. A. Medeiros for his assistance and precious suggestions.
T. N. Rabello and M. C. de Campos Vieira 317
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Tania Nunes Rabello: Departamento de Matemática, Instituto Tecnológico de Aeronáutica, São
José dos Campos, São Paulo 12228-900, Brazil
E-mail address: tania@ief.ita.br
Maria Cristina de Campos Vieira: Departamento de Matemática, Instituto Tecnológico de
Aeronáutica, São José dos Campos, São Paulo 12228-900, Brazil
E-mail address: cristina@ief.ita.br